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To find the domain of functions involving square roots, it's crucial to first solve the quadratic inequality. A quadratic inequality arises when we have a quadratic expression, like \(2x^2 - 5x + 2\), that needs to be either greater than or equal to zero. This is because a square root function requires the argument inside the root to be non-negative, otherwise it would involve imaginary numbers, which are not typically within the scope of high school algebra.

The process involves several steps:

• Start by identifying the inequality that ensures the expression inside the square root is non-negative: \(2x^2 - 5x + 2 \geq 0\).
• Set the inequality to equality to find the critical points, which are potential roots of the quadratic equation.
• Once you have the critical points, use them to divide the number line into distinct intervals.
• Test each interval to see if it satisfies the original inequality, confirming whether each section should be included in the domain.

After analyzing these intervals, the solution provides us with a domain expressed in interval notation, encompassing all valid \(x\) values that satisfy the inequality.

The square root function is represented as \(f(x) = \sqrt{g(x)}\), where \(g(x)\) must be non-negative. This is because taking the square root of negative numbers results in complex numbers, which we typically avoid in real-valued functions.

In our exercise, \(f(x) = \sqrt{2x^2 - 5x + 2}\). For this function, \(2x^2 - 5x + 2\) needs to be greater than or equal to zero to ensure the function produces real-number outputs. Every time you are dealing with square roots in the denominator or the main function, remember:

• The expression inside the square root must be non-negative
• The function's domain is directly influenced by this requirement

Converting the problem to determining when \(g(x)\) is at least zero is essential to identifying which \(x\) values keep the function well-defined and valid in the real number system.

The quadratic formula is a powerful tool to solve quadratic equations of the form \(ax^2 + bx + c = 0\). It provides a simple way to find the roots, or solutions, to the equation, using the formula:\[x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}\\]This formula is derived from completing the square of the quadratic equation and solving for \(x\). In our quadratic inequality, \(2x^2 - 5x + 2 \geq 0\), the roots are found by applying this formula with \(a = 2\), \(b = -5\), and \(c = 2\).

Solving the quadratic equation using this formula not only gives the critical points needed to test intervals but also helps determine the intervals where the expression \(2x^2 - 5x + 2\) satisfies the inequality. These roots effectively "break" the number line into analyzable sections and are essential for constructing the domain of our function. Remember:

• Check for real solutions by ensuring the discriminant \(b^2 - 4ac\) is non-negative.
• Roots indicate potential switches in the sign of the quadratic expression.